Reconstruction of MRI Data Encoded by Multiple
Nonbijective Curvilinear Magnetic Fields
Fa-Hsuan Lin,1,2* Thomas Witzel,2Gerrit Schultz,3Daniel Gallichan,3Wen-Jui Kuo,4
Fu-Nien Wang,5* Juergen Hennig,3Maxim Zaitsev,3and John W. Belliveau2
Parallel imaging technique using localized gradients (PatLoc)
uses the combination of surface gradient coils generating nonbi-
jective curvilinear magnetic fields for spatial encoding. PatLoc
imaging using one pair of multipolar spatial encoding magnetic
fields (SEMs) has two major caveats: (1) The direct inversion of
the encoding matrix requires exact determination of multiple
locations which are ambiguously encoded by the SEMs. (2)
Reconstructed images have a prominent loss of spatial resolu-
tion at the center of field-of-view using a symmetric coil array for
signal detection. This study shows that a PatLoc system actually
has a higher degree of freedom in spatial encoding to mitigate
the two challenges mentioned above. Specifically, a PatLoc sys-
tem can generate not only multipolar but also linear SEMs, which
can be used to reduce the loss of spatial resolution at the field-
of-view center. Here, we present an efficient and generalized
image reconstruction method for PatLoc imaging using multiple
SEMs without explicitly identifying the locations where SEM
encoding is not unique. Reconstructions using simulations and
empirical experimental data are compared with those using con-
ventional linear gradients to demonstrate that the general com-
bination of SEMs can improve image reconstructions.
Reson Med 68:1145–1156, 2012. V
Key words: MRI; nonlinear gradients; surface gradients;
PatLoc; parallel MRI; time-domain reconstruction
C2012 Wiley Periodicals, Inc.
MR radiofrequency (RF) coil arrays have been introduced
to generate images with high signal-to-noise ratio (SNR)
and a large field-of-view (FOV) (1). Parallel MRI was
then proposed to use distinct yet spatially correlated
data among different channels of an RF coil array to
enhance the spatiotemporal resolution of MRI at the cost
of SNR (2,3). Different from parallel MRI using highly
parallel RF detection, parallel imaging technique using
localized gradients (PatLoc) (4) uses the combination of
surface gradient coils and an RF receiver array to
improve the efficiency of MRI spatial encoding and to
reduce the peripheral nerve stimulation hazard. PatLoc
is a generalization of non-Cartesian MRI. Different from
the conventional MRI, where three spatially orthogonal
gradient systems, namely x-, y-, and z-gradients, are used
to control the magnetization precession frequency and to
encode the spatial location via frequency analysis, Pat-
Loc uses nonbijective curvilinear magnetic fields to
achieve the goal of spatial encoding. Because the mag-
uniquely encoded in a PatLoc system, parallel MRI has
to combine with a PatLoc system to accurately localize
the magnetization. Recently, prototypes of the PatLoc
system have been successfully implemented on animal
(5) and human scanners (6). So far, most PatLoc imag-
ing has focused on using one pair of multipolar spatial
encoding magnetic fields (SEMs) to encode spatial in-
formation (7–9). A reconstruction method has been pre-
sented in detail, which is fast because it allows the
application of the traditional (fast) Fourier transform.
Residual aliasing resulting from nonunique SEM encod-
ing is resolved using methods of parallel image recon-
struction. This, however, requires the explicit identifi-
cation of the multiple locations, where SEM encoding is
not unique. In generalized cases, this procedure can be
computationally intensive or even prohibitive (9). In
general, the magnetizations spatially encoded in a Pat-
Loc system can, therefore, not use traditional (fast) Fou-
rier transform to complete image reconstruction (9).
This poses a computational challenge on reconstructing
Even with the advantage of efficient image reconstruc-
tion, one caveat of using multipolar SEMs is the promi-
nent loss of the spatial resolution at the center of FOV
using a symmetric coil array. This is due to the circular
symmetry of the multipolar SEMs and the reduced sensi-
tivity of RF coils at the center of the FOV. Magnetization
distribution around the center of the FOV is similarly
encoded by both surface gradient elements and RF coils.
This ambiguity leads to a highly ill-conditioned encod-
ing and subsequently largely amplified noise around the
The purpose of this study is (1) identifying multiple
SEMs that can be realized by a generalized PatLoc
1Institute of Biomedical Engineering, National Taiwan University,
2Harvard Medical School - Athinoula A. Martinos Center for Biomedical
Imaging, Department of Radiology, Massachusetts General Hospital,
Charlestown, Massachusetts, USA.
3Department of Radiology, University Medical Center Freiburg,
4Institute of Neuroscience, National Yang-Ming University, Taipei, Taiwan.
5Department of Biomedical Engineering and Environmental Sciences,
National Tsing-Hua University, Hsinchu, Taiwan.
Part of this study has been presented in the previous Annual Meeting of
International Society of Magnetic Resonance in Medicine (ISMRM), 2009.
Grant sponsor: United States National Institutes of Health (NIH, National
R21DC010060, R21EB007298; Grant sponsor: National Science Council,
Taiwan; Grant numbers: NSC 98-2320-B-002-004-MY3, NSC 100-2325-B-
Research Institute, Taiwan; Grant number: NHRI-EX100-9715EC; Grant
sponsor: Academy of Finland (FiDiPro Program); Grant number: 127624.
Engineering, National Taiwan University, Taipei, Taiwan. E-mail: fhlin@ntu.
edu.tw or Fu-Nien Wang, Ph.D., Department of Biomedical Engineering and
Environmental Sciences, National Tsing-Hua University, Hsinchu, Taiwan.
Received 16 March 2011; revised 18 October 2011; accepted 18
Published online 13 January 2012 in Wiley Online Library (wileyonlinelibrary.
Ph.D., Institute ofBiomedical
Magnetic Resonance in Medicine 68:1145–1156 (2012)
C 2012 Wiley Periodicals, Inc.
system and (2) proposing a generalized PatLoc imaging
encoding and reconstruction using multiple SEMs to
reduce the loss of spatial resolution at the center of FOV
without explicitly identifying the multiple locations of am-
biguous SEM encoding. It has been shown that simultane-
ous switching of linear and multipolar SEMs can reduce
the loss of spatial resolution around the FOV center
(10,11). As described in the following section, we found
that the present PatLoc coil could potentially generate not
only the multipolar but also nearly linear SEMs. These
SEMs are generated by driving a linear combination of Pat-
Loc coil elements with current amplitudes and polarities
suggested by singular value decomposition (SVD). Consec-
utive application of these two types of SEMs recovers the
spatial resolution at the center of FOV. Practically, such an
image reconstruction using multiple SEMs can be effi-
ciently implemented by the conjugate gradient (CG) algo-
rithm (12). Reconstructions using simulations and experi-
mental data are compared with parallel MRI using
conventional linear gradients to illustrate that the general
combination of SEMs can improve image reconstructions.
Signal Equation and SEMs
The PatLoc image reconstruction can be theoretically
derived from the signal equation of a system using multi-
ple surface gradient elements combined with the parallel
detection across channels in a RF coil array (9):
s ¼ Er;
where s is a vector consisting of measurement across RF
coil channels using individual SEMs with a particular
driving current strength. and r is the image to be recon-
structed. Each row of the encoding matrix E represents a
spatial basis function generated from the combination of
one RF coil sensitivity profile and the time integral of a
SEM generatedfrom multiple
The simulations in this study used eight surface gradi-
ent elements with a circumferential geometry. Figure 1
shows the schematic plot of an eight-channel PatLoc sys-
tem. The magnetic field generated by each gradient ele-
ment was calculated using the Biot–Savart’s law. Given
eight magnetic fields generated by applying a unit cur-
rent on each gradient element, we used SVD to propose
USVH¼ SVDðGÞ ¼ SVDð½G1G2 ... GnG?Þ; nG¼ 8;
where Giis the spatial distribution of the z-component
of the magnetic field generated by surface gradient ele-
ment i within the FOV in a column vector. The pth col-
umn of V suggests the driving currents for gradient
elements to generate the pth SEM, whose spatial distri-
bution is described by the pth column of U. The pth di-
agonal entries of S indicate the relative proportion of the
total variance with [G1G2... GnG] by the pth SEM. Note
that SVD ensures that different SEMs are orthogonal to
each other across the FOV: UUH¼ InG, where InGis an
identity matrix of dimension nG-by-nG. Locally, different
SEMs are not necessarily orthogonal between each other.
The top five SEMs suggested by SVD are shown in Fig.
1. Interestingly, SVD automatically suggested two multi-
polar SEMs used in previous PatLoc studies (4,9). These
two are the fourth and fifth SEMs, each accounting for
6% of the total variance in G. They are named the ‘‘M1’’
and ‘‘M2’’ SEMs, respectively, in this study. The second
and third SEMs suggested by SVD have a nearly linear
magnetic field spatial distribution in two orthogonal
directions. Each SEM accounted for 21% of the total var-
iance in G each. These two SEMs are named the ‘‘L1’’
and ‘‘L2’’ and were generated by driving surface gradient
elements in a nearly sinusoidal distribution with a 90?
shift. The isointensity contours of the magnetic field gen-
erated by the first SEM are concentric rings. This field
has been used in the O-space imaging (14) and thus
named the ‘‘O’’ SEM in this study.
FIG. 1. The setup of an eight-
channel PatLoc system (upper
left). Using SVD, the five most sig-
nificant SEMs suggested by sin-
gular value decomposition (SVD)
on the magnetic fields generated
by each channel of the PatLoc
system are one O-space mode
(O), two linear modes (L1 and L2),
and two multipolar modes (M1
and M2). They constitute 42%,
21%, 21%, 6%, and 6% of the
total variance, respectively. [Color
figure can be viewed in the online
1146Lin et al.
Image Acquisitions and Reconstruction
One PatLoc imaging strategy is choosing a pair of SEMs
and then drive these two SEMs, respectively, according
to the ‘‘phase’’ and ‘‘frequency’’ encoding gradient time
tables in a typical 2D MRI pulse sequence (or two or-
thogonal ‘‘phase’’ encoding gradients in a 3D acquisition)
in a conventional MRI system (9). Thus, conventional
imaging sequences can be used directly on a PatLoc sys-
tem to acquire data. Considering a 2D PatLoc imaging
case with two provided SEMs for phase encodings,
encoded data can be mapped onto a two-dimensional
encoding space similar to the 2D k-space in the Fourier
imaging using linear gradients. As we attempted to accel-
erate PatLoc imaging, we subsampled the number of data
in the pth SEM by reducing the data samples from nE(p)
(p). The acceleration R was calculated by
Specifically, ‘‘L1’’ and ‘‘L2’’ SEMs were used together
as the first pair of two phase encoding gradients in a 3D
gradient echo sequence to collect nE(1) samples. ‘‘M1’’
and ‘‘M2’’ SEMs were used together as the second pair of
two phase encoding gradients in a 3D gradient echo
sequence to collect nE(2) samples. Without the loss of
generality, here we only investigated cases nE(1) ¼ nE(2).
In accelerated scans, we used the R1 ? R2 acceleration
rate to R1 ? R2 acceleration to represent the accelerated
acquisition of taking one sample from every consecutive
R1 pixels in the first (phase) encoding dimension and
one sample from every consecutive R2 pixels in the sec-
ond (frequency) encoding dimension. The acquired data
s from each pair of SEMs with corresponding spatial
bases (rows of E) can be vertically concatenated as
described in Eq. 1.
In this study, we investigated PatLoc imaging using (1)
a pair of multipolar ‘‘M1’’ þ ‘‘M2’’ SEMs and (2) two
pairs of nearly linear ‘‘L1’’ þ ‘‘L2’’ and multipolar ‘‘M1’’
þ ‘‘M2’’ SEMs. Accelerated PatLoc acquisitions using
aforementioned SEMs with 2-, 4-, and 8-fold accelera-
tions were also simulated for image reconstruction. To
fairly compare the reconstructed images subjected to the
same acquisition time, the number of samples in acquisi-
tions using multiple SEM pairs should be reduced com-
pared with the acquisition using only one SEM pair. For
example, using a 128 ? 128 image matrix with 8-fold
acceleration, PatLoc using only M1 þ M2 SEMs collected
2048 samples. PatLoc using M1 þ M2 þ L1 þ L2 SEMS
collected 1024 samples from the pair of M1 þ M2 SEMs
and 1024 samples from the pair of L1 þ L2 SEMs.
Because the Fourier transform can no longer be
applied to solve Eq. 1 directly due to the fact that rows
of E within each RF channel are not orthonormal in gen-
eral, we propose an iterative time-domain reconstruction
(iTDR) algorithm based on the CG method. The iTDR is a
generalization of the sensitivity encoded (SENSE) image
reconstruction with arbitrary k-space trajectories (12).
One advantage of iTDR reconstruction is that it does not
require defining locations of ambiguous SEM encoding a
procedure required by the original PatLoc reconstruc-
tions (4,9). The algorithm is schematically depicted in
Fig. 2, where I is the image intensity correction by divid-
ing the input image with sum-of-squares of the coil sen-
sitivity profiles in an RF coil array. Cidenotes the RF
coil sensitivity profile for channel i, and ACC indicates
FIG. 2. The flow chart of the iterative time-domain reconstruction (iTDR) for PatLoc reconstruction with arbitrary configuration of the
SEMs and data sampling scheme. ACC, encoding-space data subsampling; C, coil sensitivity modulation; TDR, the implementation of
Eq. 1; CG, conjugate gradient algorithm; I, the intensity correction.
Reconstructing Nonbijective Curvilinear Fields Encoded MRI 1147
subsampling the k-space data based on the pulse sequence
diagram. CG is the block of implementing the CG method
by updating the reconstruction from the previous itera-
tion. Note that the CG algorithm has been used in, for
example, parallel MRI reconstruction (12). Here, we only
used CG to solve the large linear system equation.
Simulations used high-resolution 3D T1-weighted struc-
tural MRI data. The pulse sequence was magnetization
prepared rapid gradient echo (MPRAGE) (pulse repetition
time/echo time/flip ¼ 2530 ms/3.49 ms/7?, partition
thickness ¼ 1.0 mm, matrix ¼ 256 ? 256, 256 partitions,
and FOV ¼ 256 mm ? 256 mm). One axial slice image
was selected as the input. We used the Biot–Savart’s law
to calculate the B1fields generated by an eight-channel
head coil array with geometry similar to the PatLoc sys-
tem depicted in Fig. 1. A sum-of-squares reference image
from eight channels of the RF coil array was calculated af-
ter given B1 fields. Simulations added Gaussian white
noise with SNR ¼ 1000 to avoid the overestimation of the
reconstruction quality due to the exact match between the
model and the noiseless synthetic measurements. The fi-
delity of the reconstruction was quantified by a difference
image between the reference image and the sum-of-square
image after reconstruction.
The spatial resolution of the reconstructed images was
evaluated by the point-spread functions (PSFs). Specifi-
cally, an image with all pixels with intensity zero except
a center pixel or a pixel close to the periphery of the
FOV set to one was used as the input image. The recon-
structed image was the corresponding empirical PSF of
the input image. We quantified the PSF by measuring its
full-width-half-maximum (FWHM). Recently, the con-
cept of local k-space has been proposed to examine the
spatial resolution in PatLoc imaging (13). Similar to con-
ventional MRI, k-space coordinates can be derived as the
local partial spatial derivative of the phase generated by
given SEM strength and duration k(p,q) at location r
kxðq;? rÞ ¼@
kxðq;? rÞ ¼@
where p denotes the index for different SEM, and q
denotes the index of the acquisition data for different
time integral of the SEM. However, different from con-
ventional MRI, because we have nonlinear SEMs in gen-
eral, the k-space is not identical across the FOV. Thus, it
gives the name ‘‘local k-space’’ to highlight the spatial
varying nature of Eq. 4.
We calculated the local k-space to visualize the vari-
able spatial resolution at different image voxels. All cal-
culations were implemented with MatLab (Mathworks,
Natick, MA) on a workstation with Intel 2.0 GHz Xeon
CPU and 32 GB memory.
PatLoc imaging data using linear and multipolar SEMs
were acquired on a 3-T clinical imaging system (Tim
Trio, Siemens Healthcare, Erlangen, Germany) fitted
with a custom-built gradient insert coil designed to gen-
erate two encoding fields. The geometry of each multipo-
lar SEM closely approximates a hyperbolic paraboloid,
generating two locally orthogonal SEMs rotated by 45?
with respect to each other. Details of the PatLoc arrange-
ment have been previously described in Refs. 6 and 15.
A head coil was fitted inside the gradient insert and con-
sisted of a single-channel transmitter and an eight-chan-
nel receive array (Siemens Healthcare, Erlangen, Ger-
many). Data using only traditional linear gradients and
two multipolar SEMs were acquired separately. Imaging
parameters were as follows: FOV ¼ 220 mm, image ma-
trix ¼ 256 ? 256, slice thickness ¼ 5 mm, pulse repeti-
tion time ¼ 50 ms, and echo time ¼ 8.1 ms.
Provided with SEMs and the acquisition grids, we
directly reconstructed images without explicit determi-
nation of the locations encoded ambiguously with the
SEMs using the iTDR. Figure 3 shows the convergence
behaviors of the PatLoc reconstructions using only multi-
polar SEMs (M1 þ M2) and using the combination of
multipolar and nearly linear SEMs (M1 þ M2 þ L1 þ
L2). Provided with the reference image in the synthetic
data, we calculated the percentage error of the recon-
struction at each repetition. Over the first 10 iterations,
the percentage error dropped over 80% using either M1
þ M2 or M1 þ M2 þ L1 þ L2 SEMs. The convergence
rates for different accelerated data were different: a
higher acceleration rate corresponded to a slower conver-
gence. This can be explained by the deteriorated condi-
tioning of the encoding matrix (Eq. 1) in more acceler-
ated acquisitions. We found that the reconstruction
converged after 50 iterations. For acquisitions without
acceleration (R ¼ 1 ? 1), with 2-fold (R ¼ 2 ? 1) and 4-
fold (R ¼ 2 ? 2) accelerations, the final reconstruction
has the percentage error less than 5%. Eight-fold acceler-
ation (R ¼ 2 ? 4) converged at 5 and 3% error using M1
þ M2 and M1 þ M2 þ L1 þ L2 SEMs, respectively.
Figure 4 shows the reconstructed images and the corre-
sponding error images at different iterations without
acceleration and with 8-fold (R ¼ 2 ? 4) accelerations. For
comparison, results using M1 þ M2 SEMs and M1 þ M2 þ
L1 þ L2 SEMs were shown together. Starting from a zero
image, the reconstruction started from the center of the
FOV during the first few repetitions. These images corro-
borated with the convergence plots shown in Fig. 3: a
higher acceleration rate converged more slowly. The final
reconstructed images without acceleration had a percent-
age error of 1% and with 8-fold accelerations the recon-
struction had percentage errors of 5 and 3% using M1 þ
M2 SEMs and M1 þ M2 þ L1 þ L2 SEMs, respectively.
Details of the final reconstructions are shown in Fig. 5.
For comparison, we also showed the reference images
and the reconstruction using conventional linear gra-
dients. Without acceleration (Fig. 5, top row), we found
that all reconstructions around cortex were satisfactory.
Minor reconstruction error (<1%) was due to the simu-
lated noise. However, at the center of the FOV, the Pat-
Loc reconstruction using only M1 þ M2 was blurred.
Such a loss of spatial resolution at the center of the
image (indicated by a yellow arrow head) was consistent
with previous studies (4,9): SEMs and coil sensitivity
1148Lin et al.
maps from a coil array cannot provide sufficient spatial
information to reliably resolve images in the FOV center.
Note that by using the L1 and L2 SEMs in conjunction
with the M1 and M2 SEMs, we improved the loss of the
spatial resolution around the FOV center significantly.
The reconstruction was found similar to the one from
using the conventional linear gradient system over the
Accelerated reconstructions are also shown in Fig. 5.
At 4-fold acceleration (R ¼ 2 ? 2), PatLoc reconstruction
with multipolar SEMs (M1 þ M2) still have prominent
loss of spatial information at the FOV center (yellow arrow
heads). Using linear gradient system or PatLoc with multi-
polar and nearly linear SEMs generated comparable recon-
structions to the reference image (Fig. 5, middle row). The
noise level of the reconstruction was found marginally
higher in the 4-fold accelerated case (0.9%) than the unac-
celerated case (0.8%), potentially due to the 50% reduc-
tion on the sample and/or the noise amplification during
the reconstruction. Eight-fold acceleration showed clear
differences among reconstructions (Fig. 5, bottom row).
As this PatLoc system has eight RF coils, the maximal
acceleration rate was eight before transforming the signal
equation (Eq. 1) from an overdetermined linear system
into an underdetermined linear system. Using the linear
gradient system, clear residual aliasing artifact along the
left–right direction was found in the 2 ? 4 acceleration
(green arrow heads). PatLoc reconstructions using only
multipolar SEMs (M1 þ M2) showed the loss of spatial in-
formation at the FOV center (yellow arrow heads) and
noisy reconstruction at the frontal and occipital areas (ma-
genta arrow heads). With M1, M2, L1, and L2 SEMs, the 8-
fold reconstructed PatLoc image showed less aliasing arti-
fact than the linear gradient reconstruction, improved spa-
tial resolution in the FOV center, and reduced noise level
in the frontal and occipital lobes.
Because conventional MRI readily provides highly lin-
ear gradient coils, we wondered how reconstructions
change if we replace the L1 þ L2 SEMs generated by the
PatLoc system with the two linear Bz(linear 1 þ linear 2)
generated by the conventional MRI gradient coils. Figure 6
shows such comparison without acceleration and with 4-
fold and 8-fold accelerations (R ¼ 2 ? 2 and R ¼ 2 ? 4).
We found that reconstructions using either L1 þ L2 or
linear 1 þ linear 2 are pretty similar at R ¼ 1 and R ¼ 2
? 2. At a high acceleration rate R ¼ 2 ? 4, the recon-
struction noise was more prominent using the linear Bz.
This might be due to the difficulty of using the RF sensi-
tivities to interpolate the missing spatial bases generated
by linear Bz.
Figure 7 shows the reconstructions of different SNR at R
¼ 2 ? 2 and R ¼ 2 ? 8. We can see that at a fixed accelera-
tion rate, the reconstruction deteriorated as the SNR
decreased. Notably, at SNR ¼ 100, R ¼ 2 ? 2 shows fairly
good reconstruction (residual error ¼ 0.3%). At R ¼ 2 ? 4,
the reconstruction shows noticeable noises at SNR ¼ 100
(residual error ¼ 3.4%). We consider that the reconstruc-
tion can work satisfactorily at SNR ¼ 100 or higher.
The proposed iTDR reconstruction has the capability
of reconstructing images using arbitrary SEMs. Figure 8
shows an example of using two SEMs generated by ran-
domly weighting the surface gradient elements. In this
example, details of images can be restored in lower left
corner of the FOV based on sufficient spatial information
from the highly nonlinear SEM pairs and RF coil
To evaluate the spatial resolution, PSFs corresponding
to a pixel at the FOV center and the peripheral of the
FOV are shown in Fig. 9. The locations of the nonzero
image pixel for the PSF input image are surrounded by
cyan boxes and indicated by cyan arrow heads. Although
the PSF at the FOV periphery was very focal, we found
that the center of FOV has a spatially blurred PSF when
only M1 þ M2 SEMs were used. Using L1 þ L2 SEMs
and M1 þ M2 SEMs, the PSF at the FOV periphery can
maintain focal, and the PSF at the FOV center can be
improved clearly by suppressing side lobes. The bottom
panel of Fig. 7 plots the profile of the PSF along a verti-
cal line passing through the nonzero pixel in the PSF
input image. The location of the nonzero input PSF
FIG. 3. The convergence of the iTDR PatLoc reconstructions using M1 þ M2 SEM (left) and M1 þ M2 þ L1 þ L2 SEM (right) with R ¼
1 ? 1, R ¼ 2 ? 1, R ¼ 2 ? 2, and R ¼ 2 ? 4 accelerations. Convergent reconstructions were found after 50 iterations in general. [Color
figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
Reconstructing Nonbijective Curvilinear Fields Encoded MRI 1149
image is indicated by a gray dashed line. Quantitatively,
the FWHMs at FOV periphery using M1 þ M2 SEMs and
M1 þ M2 þ L1 þ L2 SEMs were both 1.0 pixel. The
FWHMs at FOV center using M1 þ M2 SEMs and M1 þ
M2 þ L1 þ L2 SEMs were 7.0 and 2.2 pixels, respectively.
We also found that the peak of the center FOV PSF shifted
by 2 pixels when only M1 þ M2 SEMs were used.
PSFs were also evaluated for reconstructions using the
accelerated data. Figure 10 plots the profiles of the PSFs
for input images with a nonzero pixel at the FOV center
(Fig. 10, left) and at the FOV periphery (Fig. 10, right).
We found that acceleration modulated the PSF margin-
ally. Using only multipolar SEMs, the PSFs at the FOV
center were blurred with FWHMs of 7.0 pixels, 7.4 pix-
els, and 7.8 pixels for R ¼ 1 ? 1, R ¼ 2 ? 2, and R ¼ 2 ?
4, respectively. The peak of the PSF was also found
shifted by 2 pixels for all acquisitions. Using multipolar
and linear SEMs, the PSFs at the FOV center were much
focal with FWHMs of 2.2 pixels, 2.3 pixels, and 2.4 pix-
els for R ¼ 1 ? 1, R ¼ 2 ? 2, and R ¼ 2 ? 4, respectively.
At the FOV periphery, all reconstructions using either
M1 þ M2 SEMs or M1 þ M2 þ L1 þ L2 SEMs had a PSF
of 1.0 pixel for unaccelerated (R ¼ 1 ? 1) and accelerated
(R ¼ 2 ? 2 and R ¼ 2 ? 4) acquisitions.
FIG. 4. Individual reconstructed images and error images using iterative time-domain reconstruction (iTDR). Each subplot shows the
reconstruction (left) and the residual error image (right) in each repetition. At a higher acceleration rate, the convergence was slower.
[Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
1150 Lin et al.
The spatial resolution analysis using PSF corroborates
the local k-space analysis. Figure 11 shows that the local
k-space at 7 ? 7 image voxels evenly distributed over the
FOV. Using only multipolar SEMs has a low spatial reso-
lution around the FOV center, consistent with previous
studies (13). FOV periphery does not lose spatial resolu-
tion. These local k-space calculations are consistent with
the PSF calculations (Figs. 9 and 10). Using both multi-
polar and nearly linear SEMs, the local k-space shows
improved spatial resolution around the FOV center as
the result of increased k-space coverage. From the local
k-space plot, it is evident that the corresponding local k-
space distribution is the sum of local k-space using indi-
vidual SEMs, as also described in Eq. 4.
Experimental reconstructions using data with linear
gradients, multipolar SEMs (M1 þ M2), and four SEMs
together are shown in Fig. 12. As limited by currently
available hardware, the linear gradients were generated
from the linear gradient coils in the conventional MRI
system. Because our calculation (Fig. 6) shows little dif-
ference between reconstructions using nearly linear
SEMs generated by the generalized PatLoc system and
the linear gradients generated by the conventional MRI
gradient coils, we considered Fig. 12 is what can be
achieved using L1/L2 and M1/M2 SEMs by the PatLoc
system. Compared with reconstruction using only multi-
polar SEM acquisitions, the loss of spatial resolution
around the FOV center was reduced when four SEMs
were used together. However, we noticed that around
the FOV center the spatial resolution was not recovered
completely. There were also more prominent residual
aliasing artifacts when four SEMs were used. Similar to
a recently publishedstudy (13),this suboptimal
FIG. 5. Reconstructed
using conventional linear gradient
coils and PatLoc system with mul-
tipolar (M1 þ M2) SEMs and with
multipolar as well as linear (M1 þ
M2 þ L1 þ L2) SEMs. Reconstruc-
tions using unaccelerated acquisi-
acceleration (R ¼ 2 ? 2), and 8-
fold acceleration (R ¼ 2 ? 4) are
shown in the top, middle, and bot-
tom rows, respectively. PatLoc
using M1 þ M2 SEMs shows
reduced image resolution at the
FOV center (yellow arrow heads).
At 8-fold acceleration, conven-
tional linear gradients generate
strong aliasing artifacts at 8-fold
acceleration (green arrow heads)
and PatLoc with M1 þ M2 SEMs
shows noisy reconstructions at the
frontal and occipital lobes (ma-
genta arrow heads). PatLoc with
M1 þ M2 þ L1 þ L2 SEMs has
comparably better image quality.
[Color figure can be viewed in the
online issue, which is available at
FIG. 6. PatLoc imaging reconstructions using the multipolar (M1
þ M2) SEMs and the nearly linear (L1 þ L2) SEMs generated by
the PatLoc system or the multipolar (M1 þ M2) SEMs together
with the two linear (linear 1 þ linear 2) Bzfields generated by the
conventional gradient coils. [Color figure can be viewed in the
online issue, which is available at wileyonlinelibrary.com.]
Reconstructing Nonbijective Curvilinear Fields Encoded MRI 1151
performance is likely due to errors in the estimation of
multipolar SEMs and RF coil sensitivity maps. For exam-
ple, the coil sensitivity maps were estimated from a sepa-
rate reference scan. Even though such a method ensured
consistent coil loading such that estimated coil sensitivity
is consistent between the reference scan and PatLoc imag-
ing scan, motion between two scans as well as different
eddy current effects can lead to inaccurate coil sensitivity
estimation and cause imperfect reconstruction.
This study addresses two challenges of the PatLoc imag-
ing. First, we proposed the PatLoc acquisitions using
multiple (potentially arbitrary) SEMs and a generalized
image reconstruction algorithm without explicitly defin-
ing the locations identically encoded by the SEMs.
Although we only demonstrated the iTDR reconstruc-
tions using multipolar and nearly linear SEMs, the same
algorithm can be applied to different SEMs directly (Fig.
FIG. 7. Accelerated (R ¼ 2 ? 2 and R ¼ 2 ? 4) PatLoc imaging reconstructions using the multipolar (M1 þ M2) and the nearly linear
(L1 þ L2) SEMs at SNR ¼ 1000, 500, 200, 100, 50, 20, and 10. [Color figure can be viewed in the online issue, which is available at
1152 Lin et al.
8). Second, we investigated different SEMs that can be
generated from a given PatLoc system. Importantly, two
nearly linear SEMs were revealed via SVD. Previously,
using multipolar SEMs alone, the reconstructed image
has a poor spatial resolution at the center of FOV, where
the spatial information from the RF coil sensitivity pro-
files and SEMs are insufficient to localize precession
magnetization accurately (4,9). This challenge was miti-
gated by using multipolar and linear SEMs together, as
demonstrated in Figs. 4 and 5. For unaccelerated image
acquisitions, the image was found much improved in the
FOV center. For accelerated acquisitions, PatLoc recon-
structions using multipolar and linear SEMs have less
aliasing artifact than using acquisitions in a linear gradi-
ent system and reduced noise compared with using mul-
tipolar SEMs only.
It should be noted that the realized PatLoc system is
different from our simulation setup, as we do not cur-
rently have hardware in place to drive all eight elements
of the PatLoc insert coil independently. However, our
simulation is still valid as this setup has been used in
the original PatLoc study showing good agreement
between the simulations and experimental data (4). The
SEMs are also quasi-static magnetic fields, and thus
Biot–Savart’s law can be used to generate reasonable
field patterns. Different practical designs have been
described in Refs. 9,13, and 16–18.
Via SVD, we found that nearly linear SEMs can be
generated from the linear combination of surface gradient
elements directly without using the conventional linear
gradient coils. Considering multiple surface gradient
elements arranged circumferentially with a uniform
spacing, driving these elements with current amplitudes
following a single cycle sinusoid pattern can generate
fairly linear SEMs. Even though the current imaging
hardware may not have this capability, our results sug-
gest that a PatLoc system without traditional linear gra-
dients can actually do all imaging experiments in a con-
ventional MRI system depending on linear gradients
coils. Because linear gradient coils can be theoretically
replaced by PatLoc linear SEMs, a wider bore size, for
example, compared with the existing PatLoc system is
possible if the surface gradient elements can be as
powerful and efficient as the whole-body gradient sys-
tem. It should be noted that the PatLoc coil used in our
experiments was hardwired to produce the multipolar
M1- and M2-fields (Fig. 1) only. Technical realization of
a freely configurable array of eight independent coil ele-
ments may be challenging in terms of equipment setup,
eddy current behavior, and mechanical stabilization.
FIG. 8. iTDR can reconstruct images using arbitrary SEMs. Two
SEMs generated by randomly weighting the gradient elements are
shown at top. A reference image (bottom left) encoded by these
two SEMs and iTDR reconstructs the highly distorted image (bot-
tom right) efficiently. [Color figure can be viewed in the online
issue, which is available at wileyonlinelibrary.com.]
FIG. 9. The point-spread function (PSF) at the center of the FOV
(left column) and the periphery of the FOV (right column) for unac-
celerated PatLoc imaging using either multipolar (M1 þ M2) or
multipolar and linear (M1 þ M2 þ L1 þ L2) SEMs. The cyan boxes
and cyan arrow heads indicate the location of the nonzero input
image pixel in PSF evaluation. The bottom panel shows the profile
of the PSF along a vertical line passing through the nonzero PSF
input image pixel. The gray dashed line indicates the location of
the nonzero PSF input image pixel. [Color figure can be viewed in
the online issue, which is available at wileyonlinelibrary.com.]
Reconstructing Nonbijective Curvilinear Fields Encoded MRI 1153
SVD on the collection of Bzfrom all surface gradient
elements revealed not only multipolar and linear SEMs
but also the O-ring SEM and other configurations (Fig.
3). Notably, the O-ring SEM corresponded to the most
significant singular value/vector (42% of the total var-
iance). However, we want to clarify that singular values
and the associated singular vectors here are not directly
related to the spatial resolution or the reconstruction effi-
ciency of PatLoc imaging. In fact, the spatial bases used for
PatLoc imaging are spatially dependent complex sinusoids
with phases proportional to the temporal integral of SEMs.
The ‘‘O’’ SEM was indeed the SEM used in O-space imag-
ing (14). In fact, the implementation of O-space imaging
uses not only the ‘‘O’’ SEM but also the linear SEMs
offered by the conventional imaging gradients to make dif-
ferent center placements. As the generalized PatLoc system
can have the capability of generating nearly linear SEMs,
O-space imaging can be implemented on a PatLoc system.
The encoding fields from previous PatLoc and O-space
implementations are, thus, theoretically related to each
other by using different SEMs generated from a system of
surface gradient elements with a circumferential arrange-
ment. Although we did not explore including O-ring SEM
for PatLoc imaging here, some preliminary studies have
suggested that different SEMs, including O-ring, M1/M2,
and L1/L2 SEMs, can be used jointly to improve the spatial
FIG. 10. The point-spread function (PSF) at the center of the FOV (left) and the periphery of the FOV (right) for unaccelerated (R ¼ 1 ?
1) and accelerated (R ¼ 2 ? 2 and R ¼ 2 ? 4) PatLoc imaging using either multipolar (M1 þ M2) or multipolar and linear (M1 þ M2 þ
L1 þ L2) SEMs. The gray dashed line indicates the location of the nonzero PSF input image pixel. [Color figure can be viewed in the
online issue, which is available at wileyonlinelibrary.com.]
FIG. 11. The local k-space for unaccelerated PatLoc imaging using either multipolar (M1 þ M2) or multipolar and linear (M1 þ M2 þ L1
þ L2) SEMs. Improved spatial resolution around the FOV center was observed using four SEMs due to a wider k-space coverage.
[Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
1154Lin et al.
encoding efficiency (19). We will pursue this research
topic in the near future.
Generating nearly linear SEMs is the outcome of a Pat-
Loc system with eight surface gradient elements in a
symmetric arrangement. Other hardware configurations
may not be able to generate nearly linear SEMs. In cases
where nearly linear SEMs were not available, it is possi-
ble to use the linear gradient coils readily in the conven-
tional MRI system together with the PatLoc SEMs to
reduce the spatial resolution loss at the FOV center
when the interactions between two systems have been
considered carefully. This was demonstrated in Fig. 6.
This study investigated accelerated image acquisitions
and reconstructions using PatLoc SEMs (Figs. 4 and 5).
However, defining acceleration
encoding can be more complicated than just simply con-
trolling the data acquisition time. Specifically, fair com-
parisons between accelerated and unaccelerated data
become difficult due to the varying spatial resolution
across the FOV. Although the number of samples was
kept the same across all encoding schemes, the effective
local resolution was different. For example, acceleration
could be achieved by acquiring fully sampled low-reso-
images may be complicated by residual aliasing artifacts.
Related to the heterogeneous spatial resolution in Pat-
depended on the object used for simulation. If the object
had fairly low spatial resolution, then the errors would
The results reported in this study can be further gener-
alized in a few directions. First, the acquisition grid was
only limited to the uniform sampling case. It is possible
to adopt sampling patterns with spatially heterogeneous
density in the encoding space. Considering natural
images with dominant low spatial frequency components
in general, it is reasonable to oversample the encoding
space corresponding to spatial bases showing slow spa-
tial variability to obtain better results. The iTDR recon-
struction immediately allows such image reconstruc-
tions. This is different from reconstruction based on the
fast Fourier transform, which usually requires data sam-
ples separated by equal spacing in the encoding space.
The iTDR reconstruction can avoid the necessary regrid-
ding procedure from a non-Cartesian grid to a Cartesian
grid and, in theory, calculate the reconstruction from ar-
bitrary sampling patterns.
Encoding using nearly linear and multipolar SEMs can
be realized in different approaches. This study proposed
reportedin this study
the acquisitions of using multipolar SEM pair and nearly
linear SEM pair separately. Note that it has been recently
suggested that multipolar and linear SEMs can be simul-
taneously turned on to achieve similar results of a more
homogeneous spatial resolution (13). Differently, the
approach suggested in this study is to turn on multipolar
and linear SEMs consecutively. Here, we only investi-
gated the case where samples from each multipolar and
linear SEM pairs constitute 50% of the total data sam-
ples. It is possible to release from such a constraint that
different SEM pairs acquire different proportions of sam-
ples such that the acquisition and the reconstruction can
be further optimized.
Even though we used SVD to reveal combinations of
driving currents on the surface gradient elements to gen-
erate SEMs, it is important to notice that using alterna-
tive decompositions can reveal other potentially interest-
ing SEMs. SVD ensures that different SEMs are globally
orthogonal with each other. Such an othogonality condi-
tion, in fact, may not be required in the generalized MRI
using nonlinear SEMs. Elucidating the optimal SEMs
remains an open question.
In this study, we spatially encode the object by
pairs of SEMs. Provided with nSEM SEMs, there exists
nSEM(nSEM? 1)/2 choices of SEM pairs, each of which can
manipulate its strength to acquire a data sample onto a 2D
encoding space. More generally, it is actually possible to
modulate the strength of nSEMSEMs simultaneously. This
is effectively encoding an object onto an nSEM-dimen-
sional encoding space. Such an increase in the encoding
dimensionality brings more degrees of freedom in the spa-
tial basis functions generated from the conjunction of gra-
dient elements and RF coil. Hypothetically, we may more
efficiently encode the object: images can be more accu-
rately described using fewer spatial bases. This will be fur-
ther investigated in the near future.
PatLoc potentially can be used in applications where a
high slew rate is needed with reduced nerve stimulation
hazard (4). Additionally, it has been shown that PatLoc
system can be used to achieve small FOV imaging (20)
and parallel transmission (21,22). However, the price to
pay includes the complexity in hardware availability,
control, and image reconstruction. Advantages and dis-
advantages should be traded-off by considering the na-
ture of different MR experiments.
The authors acknowledge Anna Masako Welz and Chris A.
Cocosco for their contribution to the experimental setup in
Freiburg and support of in vivo measurements.
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